The Competition Complexity of Prophet Inequalities with Correlations

TL;DR

This paper explores how correlations among rewards affect the resource augmentation needed for prophet inequalities, revealing that correlations can significantly increase the complexity and requiring new algorithms for optimal approximation.

Contribution

It extends prophet inequality analysis to correlated rewards, showing the dependence on the number of rewards and developing asymptotically optimal algorithms for complex scenarios.

Findings

The required number of additional rewards depends on the original reward count.

Block-threshold algorithms may need infinitely many rewards under correlations.

New algorithms are optimal for specific correlated reward scenarios.

Abstract

We initiate the study of the prophet inequality problem through the resource augmentation framework in scenarios when the values of the rewards are correlated. Our goal is to determine the number of additional rewards an online algorithm requires to approximate the maximum value of the original instance. While the independent reward case is well understood, we extend this research to account for correlations among rewards. Our results demonstrate that, unlike in the independent case, the required number of additional rewards for approximation depends on the number of original rewards, and that block-threshold algorithms, which are optimal in the independent case, may require an infinite number of additional rewards when correlations are present. We develop asymptotically optimal algorithms for the following three scenarios: (1) where rewards arrive in blocks corresponding to the different copies of the original instance; (2) where rewards across all copies are arbitrarily shuffled; and (3) where rewards arrive in blocks corresponding to the different copies of the original instance, and values within each block are pairwise independent rather than fully correlated.